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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 273273.bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 273273.bo1 | 273273bo4 | \([1, 1, 0, -56840956, -164969072939]\) | \(35765103905346817/1287\) | \(730847727376767\) | \([2]\) | \(16515072\) | \(2.7965\) | |
| 273273.bo2 | 273273bo5 | \([1, 1, 0, -24917701, 46348739494]\) | \(3013001140430737/108679952667\) | \(61716003432860573143347\) | \([2]\) | \(33030144\) | \(3.1431\) | |
| 273273.bo3 | 273273bo3 | \([1, 1, 0, -3925366, -2005004945]\) | \(11779205551777/3763454409\) | \(2137150040329233698769\) | \([2, 2]\) | \(16515072\) | \(2.7965\) | |
| 273273.bo4 | 273273bo2 | \([1, 1, 0, -3552721, -2578505600]\) | \(8732907467857/1656369\) | \(940601025133899129\) | \([2, 2]\) | \(8257536\) | \(2.4500\) | |
| 273273.bo5 | 273273bo1 | \([1, 1, 0, -198916, -49065869]\) | \(-1532808577/938223\) | \(-532787993257663143\) | \([2]\) | \(4128768\) | \(2.1034\) | \(\Gamma_0(N)\)-optimal |
| 273273.bo6 | 273273bo6 | \([1, 1, 0, 11104649, -13647254564]\) | \(266679605718863/296110251723\) | \(-168151907167612241516643\) | \([2]\) | \(33030144\) | \(3.1431\) |
Rank
sage: E.rank()
The elliptic curves in class 273273.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 273273.bo do not have complex multiplication.Modular form 273273.2.a.bo
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
